| L(s) = 1 | − 1.28·7-s − 4.14·11-s + 6.52·13-s + 5.98·17-s + 7.17·19-s − 7.53·23-s − 5.19·29-s + 5.17·31-s + 5.24·37-s − 0.680·41-s − 1.28·43-s + 5.31·47-s − 5.35·49-s + 2.21·53-s + 7.60·59-s − 2.17·61-s − 15.6·67-s − 5.50·71-s − 7.80·73-s + 5.31·77-s + 6·79-s − 9.75·83-s + 10.0·89-s − 8.35·91-s + 14.3·97-s + 0.680·101-s − 2.56·103-s + ⋯ |
| L(s) = 1 | − 0.484·7-s − 1.24·11-s + 1.80·13-s + 1.45·17-s + 1.64·19-s − 1.57·23-s − 0.964·29-s + 0.930·31-s + 0.861·37-s − 0.106·41-s − 0.195·43-s + 0.774·47-s − 0.765·49-s + 0.304·53-s + 0.990·59-s − 0.278·61-s − 1.90·67-s − 0.653·71-s − 0.913·73-s + 0.605·77-s + 0.675·79-s − 1.07·83-s + 1.06·89-s − 0.876·91-s + 1.45·97-s + 0.0677·101-s − 0.252·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.988288850\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.988288850\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 1.28T + 7T^{2} \) |
| 11 | \( 1 + 4.14T + 11T^{2} \) |
| 13 | \( 1 - 6.52T + 13T^{2} \) |
| 17 | \( 1 - 5.98T + 17T^{2} \) |
| 19 | \( 1 - 7.17T + 19T^{2} \) |
| 23 | \( 1 + 7.53T + 23T^{2} \) |
| 29 | \( 1 + 5.19T + 29T^{2} \) |
| 31 | \( 1 - 5.17T + 31T^{2} \) |
| 37 | \( 1 - 5.24T + 37T^{2} \) |
| 41 | \( 1 + 0.680T + 41T^{2} \) |
| 43 | \( 1 + 1.28T + 43T^{2} \) |
| 47 | \( 1 - 5.31T + 47T^{2} \) |
| 53 | \( 1 - 2.21T + 53T^{2} \) |
| 59 | \( 1 - 7.60T + 59T^{2} \) |
| 61 | \( 1 + 2.17T + 61T^{2} \) |
| 67 | \( 1 + 15.6T + 67T^{2} \) |
| 71 | \( 1 + 5.50T + 71T^{2} \) |
| 73 | \( 1 + 7.80T + 73T^{2} \) |
| 79 | \( 1 - 6T + 79T^{2} \) |
| 83 | \( 1 + 9.75T + 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.76715879741981512004143848019, −7.37807141545067297376290837326, −6.15838809867181516406189013548, −5.86011195884456955968179936549, −5.21982033375412228088402830787, −4.15307300181887004694307978981, −3.39287719956716623984173546838, −2.89861087751840680271242942769, −1.66765527171259496989546726358, −0.72008104256037906846306939847,
0.72008104256037906846306939847, 1.66765527171259496989546726358, 2.89861087751840680271242942769, 3.39287719956716623984173546838, 4.15307300181887004694307978981, 5.21982033375412228088402830787, 5.86011195884456955968179936549, 6.15838809867181516406189013548, 7.37807141545067297376290837326, 7.76715879741981512004143848019
